author | wenzelm |
Sun, 21 Mar 2010 17:28:35 +0100 | |
changeset 35850 | dd2636f0f608 |
parent 35849 | b5522b51cb1e |
child 41413 | 64cd30d6b0b8 |
permissions | -rw-r--r-- |
35849 | 1 |
(* Title: HOL/Algebra/Group.thy |
2 |
Author: Clemens Ballarin, started 4 February 2003 |
|
13813 | 3 |
|
4 |
Based on work by Florian Kammueller, L C Paulson and Markus Wenzel. |
|
5 |
*) |
|
6 |
||
28823 | 7 |
theory Group |
8 |
imports Lattice FuncSet |
|
9 |
begin |
|
13813 | 10 |
|
14963 | 11 |
section {* Monoids and Groups *} |
13936 | 12 |
|
20318
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
19984
diff
changeset
|
13 |
subsection {* Definitions *} |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
19984
diff
changeset
|
14 |
|
13813 | 15 |
text {* |
14963 | 16 |
Definitions follow \cite{Jacobson:1985}. |
13813 | 17 |
*} |
18 |
||
14963 | 19 |
record 'a monoid = "'a partial_object" + |
20 |
mult :: "['a, 'a] \<Rightarrow> 'a" (infixl "\<otimes>\<index>" 70) |
|
21 |
one :: 'a ("\<one>\<index>") |
|
13817 | 22 |
|
35847 | 23 |
definition |
14852 | 24 |
m_inv :: "('a, 'b) monoid_scheme => 'a => 'a" ("inv\<index> _" [81] 80) |
35848
5443079512ea
slightly more uniform definitions -- eliminated old-style meta-equality;
wenzelm
parents:
35847
diff
changeset
|
25 |
where "inv\<^bsub>G\<^esub> x = (THE y. y \<in> carrier G & x \<otimes>\<^bsub>G\<^esub> y = \<one>\<^bsub>G\<^esub> & y \<otimes>\<^bsub>G\<^esub> x = \<one>\<^bsub>G\<^esub>)" |
13936 | 26 |
|
35847 | 27 |
definition |
14651 | 28 |
Units :: "_ => 'a set" |
14852 | 29 |
--{*The set of invertible elements*} |
35848
5443079512ea
slightly more uniform definitions -- eliminated old-style meta-equality;
wenzelm
parents:
35847
diff
changeset
|
30 |
where "Units G = {y. y \<in> carrier G & (\<exists>x \<in> carrier G. x \<otimes>\<^bsub>G\<^esub> y = \<one>\<^bsub>G\<^esub> & y \<otimes>\<^bsub>G\<^esub> x = \<one>\<^bsub>G\<^esub>)}" |
13936 | 31 |
|
32 |
consts |
|
35850 | 33 |
pow :: "[('a, 'm) monoid_scheme, 'a, 'b::number] => 'a" (infixr "'(^')\<index>" 75) |
34 |
||
35 |
overloading nat_pow == "pow :: [_, 'a, nat] => 'a" |
|
36 |
begin |
|
37 |
definition "nat_pow G a n = nat_rec \<one>\<^bsub>G\<^esub> (%u b. b \<otimes>\<^bsub>G\<^esub> a) n" |
|
38 |
end |
|
13936 | 39 |
|
35850 | 40 |
overloading int_pow == "pow :: [_, 'a, int] => 'a" |
41 |
begin |
|
42 |
definition "int_pow G a z = |
|
43 |
(let p = nat_rec \<one>\<^bsub>G\<^esub> (%u b. b \<otimes>\<^bsub>G\<^esub> a) |
|
44 |
in if neg z then inv\<^bsub>G\<^esub> (p (nat (-z))) else p (nat z))" |
|
45 |
end |
|
13813 | 46 |
|
19783 | 47 |
locale monoid = |
48 |
fixes G (structure) |
|
13813 | 49 |
assumes m_closed [intro, simp]: |
14963 | 50 |
"\<lbrakk>x \<in> carrier G; y \<in> carrier G\<rbrakk> \<Longrightarrow> x \<otimes> y \<in> carrier G" |
51 |
and m_assoc: |
|
52 |
"\<lbrakk>x \<in> carrier G; y \<in> carrier G; z \<in> carrier G\<rbrakk> |
|
53 |
\<Longrightarrow> (x \<otimes> y) \<otimes> z = x \<otimes> (y \<otimes> z)" |
|
54 |
and one_closed [intro, simp]: "\<one> \<in> carrier G" |
|
55 |
and l_one [simp]: "x \<in> carrier G \<Longrightarrow> \<one> \<otimes> x = x" |
|
56 |
and r_one [simp]: "x \<in> carrier G \<Longrightarrow> x \<otimes> \<one> = x" |
|
13817 | 57 |
|
13936 | 58 |
lemma monoidI: |
19783 | 59 |
fixes G (structure) |
13936 | 60 |
assumes m_closed: |
14693 | 61 |
"!!x y. [| x \<in> carrier G; y \<in> carrier G |] ==> x \<otimes> y \<in> carrier G" |
62 |
and one_closed: "\<one> \<in> carrier G" |
|
13936 | 63 |
and m_assoc: |
64 |
"!!x y z. [| x \<in> carrier G; y \<in> carrier G; z \<in> carrier G |] ==> |
|
14693 | 65 |
(x \<otimes> y) \<otimes> z = x \<otimes> (y \<otimes> z)" |
66 |
and l_one: "!!x. x \<in> carrier G ==> \<one> \<otimes> x = x" |
|
67 |
and r_one: "!!x. x \<in> carrier G ==> x \<otimes> \<one> = x" |
|
13936 | 68 |
shows "monoid G" |
27714
27b4d7c01f8b
Tuned (for the sake of a meaningless log entry).
ballarin
parents:
27713
diff
changeset
|
69 |
by (fast intro!: monoid.intro intro: assms) |
13936 | 70 |
|
71 |
lemma (in monoid) Units_closed [dest]: |
|
72 |
"x \<in> Units G ==> x \<in> carrier G" |
|
73 |
by (unfold Units_def) fast |
|
74 |
||
75 |
lemma (in monoid) inv_unique: |
|
14693 | 76 |
assumes eq: "y \<otimes> x = \<one>" "x \<otimes> y' = \<one>" |
77 |
and G: "x \<in> carrier G" "y \<in> carrier G" "y' \<in> carrier G" |
|
13936 | 78 |
shows "y = y'" |
79 |
proof - |
|
80 |
from G eq have "y = y \<otimes> (x \<otimes> y')" by simp |
|
81 |
also from G have "... = (y \<otimes> x) \<otimes> y'" by (simp add: m_assoc) |
|
82 |
also from G eq have "... = y'" by simp |
|
83 |
finally show ?thesis . |
|
84 |
qed |
|
85 |
||
27698 | 86 |
lemma (in monoid) Units_m_closed [intro, simp]: |
87 |
assumes x: "x \<in> Units G" and y: "y \<in> Units G" |
|
88 |
shows "x \<otimes> y \<in> Units G" |
|
89 |
proof - |
|
90 |
from x obtain x' where x: "x \<in> carrier G" "x' \<in> carrier G" and xinv: "x \<otimes> x' = \<one>" "x' \<otimes> x = \<one>" |
|
91 |
unfolding Units_def by fast |
|
92 |
from y obtain y' where y: "y \<in> carrier G" "y' \<in> carrier G" and yinv: "y \<otimes> y' = \<one>" "y' \<otimes> y = \<one>" |
|
93 |
unfolding Units_def by fast |
|
94 |
from x y xinv yinv have "y' \<otimes> (x' \<otimes> x) \<otimes> y = \<one>" by simp |
|
95 |
moreover from x y xinv yinv have "x \<otimes> (y \<otimes> y') \<otimes> x' = \<one>" by simp |
|
96 |
moreover note x y |
|
97 |
ultimately show ?thesis unfolding Units_def |
|
98 |
-- "Must avoid premature use of @{text hyp_subst_tac}." |
|
99 |
apply (rule_tac CollectI) |
|
100 |
apply (rule) |
|
101 |
apply (fast) |
|
102 |
apply (rule bexI [where x = "y' \<otimes> x'"]) |
|
103 |
apply (auto simp: m_assoc) |
|
104 |
done |
|
105 |
qed |
|
106 |
||
13940 | 107 |
lemma (in monoid) Units_one_closed [intro, simp]: |
108 |
"\<one> \<in> Units G" |
|
109 |
by (unfold Units_def) auto |
|
110 |
||
13936 | 111 |
lemma (in monoid) Units_inv_closed [intro, simp]: |
112 |
"x \<in> Units G ==> inv x \<in> carrier G" |
|
13943 | 113 |
apply (unfold Units_def m_inv_def, auto) |
13936 | 114 |
apply (rule theI2, fast) |
13943 | 115 |
apply (fast intro: inv_unique, fast) |
13936 | 116 |
done |
117 |
||
19981 | 118 |
lemma (in monoid) Units_l_inv_ex: |
119 |
"x \<in> Units G ==> \<exists>y \<in> carrier G. y \<otimes> x = \<one>" |
|
120 |
by (unfold Units_def) auto |
|
121 |
||
122 |
lemma (in monoid) Units_r_inv_ex: |
|
123 |
"x \<in> Units G ==> \<exists>y \<in> carrier G. x \<otimes> y = \<one>" |
|
124 |
by (unfold Units_def) auto |
|
125 |
||
27698 | 126 |
lemma (in monoid) Units_l_inv [simp]: |
13936 | 127 |
"x \<in> Units G ==> inv x \<otimes> x = \<one>" |
13943 | 128 |
apply (unfold Units_def m_inv_def, auto) |
13936 | 129 |
apply (rule theI2, fast) |
13943 | 130 |
apply (fast intro: inv_unique, fast) |
13936 | 131 |
done |
132 |
||
27698 | 133 |
lemma (in monoid) Units_r_inv [simp]: |
13936 | 134 |
"x \<in> Units G ==> x \<otimes> inv x = \<one>" |
13943 | 135 |
apply (unfold Units_def m_inv_def, auto) |
13936 | 136 |
apply (rule theI2, fast) |
13943 | 137 |
apply (fast intro: inv_unique, fast) |
13936 | 138 |
done |
139 |
||
140 |
lemma (in monoid) Units_inv_Units [intro, simp]: |
|
141 |
"x \<in> Units G ==> inv x \<in> Units G" |
|
142 |
proof - |
|
143 |
assume x: "x \<in> Units G" |
|
144 |
show "inv x \<in> Units G" |
|
145 |
by (auto simp add: Units_def |
|
146 |
intro: Units_l_inv Units_r_inv x Units_closed [OF x]) |
|
147 |
qed |
|
148 |
||
149 |
lemma (in monoid) Units_l_cancel [simp]: |
|
150 |
"[| x \<in> Units G; y \<in> carrier G; z \<in> carrier G |] ==> |
|
151 |
(x \<otimes> y = x \<otimes> z) = (y = z)" |
|
152 |
proof |
|
153 |
assume eq: "x \<otimes> y = x \<otimes> z" |
|
14693 | 154 |
and G: "x \<in> Units G" "y \<in> carrier G" "z \<in> carrier G" |
13936 | 155 |
then have "(inv x \<otimes> x) \<otimes> y = (inv x \<otimes> x) \<otimes> z" |
27698 | 156 |
by (simp add: m_assoc Units_closed del: Units_l_inv) |
13936 | 157 |
with G show "y = z" by (simp add: Units_l_inv) |
158 |
next |
|
159 |
assume eq: "y = z" |
|
14693 | 160 |
and G: "x \<in> Units G" "y \<in> carrier G" "z \<in> carrier G" |
13936 | 161 |
then show "x \<otimes> y = x \<otimes> z" by simp |
162 |
qed |
|
163 |
||
164 |
lemma (in monoid) Units_inv_inv [simp]: |
|
165 |
"x \<in> Units G ==> inv (inv x) = x" |
|
166 |
proof - |
|
167 |
assume x: "x \<in> Units G" |
|
27698 | 168 |
then have "inv x \<otimes> inv (inv x) = inv x \<otimes> x" by simp |
169 |
with x show ?thesis by (simp add: Units_closed del: Units_l_inv Units_r_inv) |
|
13936 | 170 |
qed |
171 |
||
172 |
lemma (in monoid) inv_inj_on_Units: |
|
173 |
"inj_on (m_inv G) (Units G)" |
|
174 |
proof (rule inj_onI) |
|
175 |
fix x y |
|
14693 | 176 |
assume G: "x \<in> Units G" "y \<in> Units G" and eq: "inv x = inv y" |
13936 | 177 |
then have "inv (inv x) = inv (inv y)" by simp |
178 |
with G show "x = y" by simp |
|
179 |
qed |
|
180 |
||
13940 | 181 |
lemma (in monoid) Units_inv_comm: |
182 |
assumes inv: "x \<otimes> y = \<one>" |
|
14693 | 183 |
and G: "x \<in> Units G" "y \<in> Units G" |
13940 | 184 |
shows "y \<otimes> x = \<one>" |
185 |
proof - |
|
186 |
from G have "x \<otimes> y \<otimes> x = x \<otimes> \<one>" by (auto simp add: inv Units_closed) |
|
187 |
with G show ?thesis by (simp del: r_one add: m_assoc Units_closed) |
|
188 |
qed |
|
189 |
||
13936 | 190 |
text {* Power *} |
191 |
||
192 |
lemma (in monoid) nat_pow_closed [intro, simp]: |
|
193 |
"x \<in> carrier G ==> x (^) (n::nat) \<in> carrier G" |
|
194 |
by (induct n) (simp_all add: nat_pow_def) |
|
195 |
||
196 |
lemma (in monoid) nat_pow_0 [simp]: |
|
197 |
"x (^) (0::nat) = \<one>" |
|
198 |
by (simp add: nat_pow_def) |
|
199 |
||
200 |
lemma (in monoid) nat_pow_Suc [simp]: |
|
201 |
"x (^) (Suc n) = x (^) n \<otimes> x" |
|
202 |
by (simp add: nat_pow_def) |
|
203 |
||
204 |
lemma (in monoid) nat_pow_one [simp]: |
|
205 |
"\<one> (^) (n::nat) = \<one>" |
|
206 |
by (induct n) simp_all |
|
207 |
||
208 |
lemma (in monoid) nat_pow_mult: |
|
209 |
"x \<in> carrier G ==> x (^) (n::nat) \<otimes> x (^) m = x (^) (n + m)" |
|
210 |
by (induct m) (simp_all add: m_assoc [THEN sym]) |
|
211 |
||
212 |
lemma (in monoid) nat_pow_pow: |
|
213 |
"x \<in> carrier G ==> (x (^) n) (^) m = x (^) (n * m::nat)" |
|
214 |
by (induct m) (simp, simp add: nat_pow_mult add_commute) |
|
215 |
||
27698 | 216 |
|
217 |
(* Jacobson defines submonoid here. *) |
|
218 |
(* Jacobson defines the order of a monoid here. *) |
|
219 |
||
220 |
||
221 |
subsection {* Groups *} |
|
222 |
||
13936 | 223 |
text {* |
224 |
A group is a monoid all of whose elements are invertible. |
|
225 |
*} |
|
226 |
||
227 |
locale group = monoid + |
|
228 |
assumes Units: "carrier G <= Units G" |
|
229 |
||
26199 | 230 |
lemma (in group) is_group: "group G" by (rule group_axioms) |
14761 | 231 |
|
13936 | 232 |
theorem groupI: |
19783 | 233 |
fixes G (structure) |
13936 | 234 |
assumes m_closed [simp]: |
14693 | 235 |
"!!x y. [| x \<in> carrier G; y \<in> carrier G |] ==> x \<otimes> y \<in> carrier G" |
236 |
and one_closed [simp]: "\<one> \<in> carrier G" |
|
13936 | 237 |
and m_assoc: |
238 |
"!!x y z. [| x \<in> carrier G; y \<in> carrier G; z \<in> carrier G |] ==> |
|
14693 | 239 |
(x \<otimes> y) \<otimes> z = x \<otimes> (y \<otimes> z)" |
240 |
and l_one [simp]: "!!x. x \<in> carrier G ==> \<one> \<otimes> x = x" |
|
14963 | 241 |
and l_inv_ex: "!!x. x \<in> carrier G ==> \<exists>y \<in> carrier G. y \<otimes> x = \<one>" |
13936 | 242 |
shows "group G" |
243 |
proof - |
|
244 |
have l_cancel [simp]: |
|
245 |
"!!x y z. [| x \<in> carrier G; y \<in> carrier G; z \<in> carrier G |] ==> |
|
14693 | 246 |
(x \<otimes> y = x \<otimes> z) = (y = z)" |
13936 | 247 |
proof |
248 |
fix x y z |
|
14693 | 249 |
assume eq: "x \<otimes> y = x \<otimes> z" |
250 |
and G: "x \<in> carrier G" "y \<in> carrier G" "z \<in> carrier G" |
|
13936 | 251 |
with l_inv_ex obtain x_inv where xG: "x_inv \<in> carrier G" |
14693 | 252 |
and l_inv: "x_inv \<otimes> x = \<one>" by fast |
253 |
from G eq xG have "(x_inv \<otimes> x) \<otimes> y = (x_inv \<otimes> x) \<otimes> z" |
|
13936 | 254 |
by (simp add: m_assoc) |
255 |
with G show "y = z" by (simp add: l_inv) |
|
256 |
next |
|
257 |
fix x y z |
|
258 |
assume eq: "y = z" |
|
14693 | 259 |
and G: "x \<in> carrier G" "y \<in> carrier G" "z \<in> carrier G" |
260 |
then show "x \<otimes> y = x \<otimes> z" by simp |
|
13936 | 261 |
qed |
262 |
have r_one: |
|
14693 | 263 |
"!!x. x \<in> carrier G ==> x \<otimes> \<one> = x" |
13936 | 264 |
proof - |
265 |
fix x |
|
266 |
assume x: "x \<in> carrier G" |
|
267 |
with l_inv_ex obtain x_inv where xG: "x_inv \<in> carrier G" |
|
14693 | 268 |
and l_inv: "x_inv \<otimes> x = \<one>" by fast |
269 |
from x xG have "x_inv \<otimes> (x \<otimes> \<one>) = x_inv \<otimes> x" |
|
13936 | 270 |
by (simp add: m_assoc [symmetric] l_inv) |
14693 | 271 |
with x xG show "x \<otimes> \<one> = x" by simp |
13936 | 272 |
qed |
273 |
have inv_ex: |
|
14963 | 274 |
"!!x. x \<in> carrier G ==> \<exists>y \<in> carrier G. y \<otimes> x = \<one> & x \<otimes> y = \<one>" |
13936 | 275 |
proof - |
276 |
fix x |
|
277 |
assume x: "x \<in> carrier G" |
|
278 |
with l_inv_ex obtain y where y: "y \<in> carrier G" |
|
14693 | 279 |
and l_inv: "y \<otimes> x = \<one>" by fast |
280 |
from x y have "y \<otimes> (x \<otimes> y) = y \<otimes> \<one>" |
|
13936 | 281 |
by (simp add: m_assoc [symmetric] l_inv r_one) |
14693 | 282 |
with x y have r_inv: "x \<otimes> y = \<one>" |
13936 | 283 |
by simp |
14963 | 284 |
from x y show "\<exists>y \<in> carrier G. y \<otimes> x = \<one> & x \<otimes> y = \<one>" |
13936 | 285 |
by (fast intro: l_inv r_inv) |
286 |
qed |
|
287 |
then have carrier_subset_Units: "carrier G <= Units G" |
|
288 |
by (unfold Units_def) fast |
|
28823 | 289 |
show ?thesis proof qed (auto simp: r_one m_assoc carrier_subset_Units) |
13936 | 290 |
qed |
291 |
||
27698 | 292 |
lemma (in monoid) group_l_invI: |
13936 | 293 |
assumes l_inv_ex: |
14963 | 294 |
"!!x. x \<in> carrier G ==> \<exists>y \<in> carrier G. y \<otimes> x = \<one>" |
13936 | 295 |
shows "group G" |
296 |
by (rule groupI) (auto intro: m_assoc l_inv_ex) |
|
297 |
||
298 |
lemma (in group) Units_eq [simp]: |
|
299 |
"Units G = carrier G" |
|
300 |
proof |
|
301 |
show "Units G <= carrier G" by fast |
|
302 |
next |
|
303 |
show "carrier G <= Units G" by (rule Units) |
|
304 |
qed |
|
305 |
||
306 |
lemma (in group) inv_closed [intro, simp]: |
|
307 |
"x \<in> carrier G ==> inv x \<in> carrier G" |
|
308 |
using Units_inv_closed by simp |
|
309 |
||
19981 | 310 |
lemma (in group) l_inv_ex [simp]: |
311 |
"x \<in> carrier G ==> \<exists>y \<in> carrier G. y \<otimes> x = \<one>" |
|
312 |
using Units_l_inv_ex by simp |
|
313 |
||
314 |
lemma (in group) r_inv_ex [simp]: |
|
315 |
"x \<in> carrier G ==> \<exists>y \<in> carrier G. x \<otimes> y = \<one>" |
|
316 |
using Units_r_inv_ex by simp |
|
317 |
||
14963 | 318 |
lemma (in group) l_inv [simp]: |
13936 | 319 |
"x \<in> carrier G ==> inv x \<otimes> x = \<one>" |
320 |
using Units_l_inv by simp |
|
13813 | 321 |
|
20318
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
19984
diff
changeset
|
322 |
|
13813 | 323 |
subsection {* Cancellation Laws and Basic Properties *} |
324 |
||
325 |
lemma (in group) l_cancel [simp]: |
|
326 |
"[| x \<in> carrier G; y \<in> carrier G; z \<in> carrier G |] ==> |
|
327 |
(x \<otimes> y = x \<otimes> z) = (y = z)" |
|
13936 | 328 |
using Units_l_inv by simp |
13940 | 329 |
|
14963 | 330 |
lemma (in group) r_inv [simp]: |
13813 | 331 |
"x \<in> carrier G ==> x \<otimes> inv x = \<one>" |
332 |
proof - |
|
333 |
assume x: "x \<in> carrier G" |
|
334 |
then have "inv x \<otimes> (x \<otimes> inv x) = inv x \<otimes> \<one>" |
|
335 |
by (simp add: m_assoc [symmetric] l_inv) |
|
336 |
with x show ?thesis by (simp del: r_one) |
|
337 |
qed |
|
338 |
||
339 |
lemma (in group) r_cancel [simp]: |
|
340 |
"[| x \<in> carrier G; y \<in> carrier G; z \<in> carrier G |] ==> |
|
341 |
(y \<otimes> x = z \<otimes> x) = (y = z)" |
|
342 |
proof |
|
343 |
assume eq: "y \<otimes> x = z \<otimes> x" |
|
14693 | 344 |
and G: "x \<in> carrier G" "y \<in> carrier G" "z \<in> carrier G" |
13813 | 345 |
then have "y \<otimes> (x \<otimes> inv x) = z \<otimes> (x \<otimes> inv x)" |
27698 | 346 |
by (simp add: m_assoc [symmetric] del: r_inv Units_r_inv) |
14963 | 347 |
with G show "y = z" by simp |
13813 | 348 |
next |
349 |
assume eq: "y = z" |
|
14693 | 350 |
and G: "x \<in> carrier G" "y \<in> carrier G" "z \<in> carrier G" |
13813 | 351 |
then show "y \<otimes> x = z \<otimes> x" by simp |
352 |
qed |
|
353 |
||
13854
91c9ab25fece
First distributed version of Group and Ring theory.
ballarin
parents:
13835
diff
changeset
|
354 |
lemma (in group) inv_one [simp]: |
91c9ab25fece
First distributed version of Group and Ring theory.
ballarin
parents:
13835
diff
changeset
|
355 |
"inv \<one> = \<one>" |
91c9ab25fece
First distributed version of Group and Ring theory.
ballarin
parents:
13835
diff
changeset
|
356 |
proof - |
27698 | 357 |
have "inv \<one> = \<one> \<otimes> (inv \<one>)" by (simp del: r_inv Units_r_inv) |
14963 | 358 |
moreover have "... = \<one>" by simp |
13854
91c9ab25fece
First distributed version of Group and Ring theory.
ballarin
parents:
13835
diff
changeset
|
359 |
finally show ?thesis . |
91c9ab25fece
First distributed version of Group and Ring theory.
ballarin
parents:
13835
diff
changeset
|
360 |
qed |
91c9ab25fece
First distributed version of Group and Ring theory.
ballarin
parents:
13835
diff
changeset
|
361 |
|
13813 | 362 |
lemma (in group) inv_inv [simp]: |
363 |
"x \<in> carrier G ==> inv (inv x) = x" |
|
13936 | 364 |
using Units_inv_inv by simp |
365 |
||
366 |
lemma (in group) inv_inj: |
|
367 |
"inj_on (m_inv G) (carrier G)" |
|
368 |
using inv_inj_on_Units by simp |
|
13813 | 369 |
|
13854
91c9ab25fece
First distributed version of Group and Ring theory.
ballarin
parents:
13835
diff
changeset
|
370 |
lemma (in group) inv_mult_group: |
13813 | 371 |
"[| x \<in> carrier G; y \<in> carrier G |] ==> inv (x \<otimes> y) = inv y \<otimes> inv x" |
372 |
proof - |
|
14693 | 373 |
assume G: "x \<in> carrier G" "y \<in> carrier G" |
13813 | 374 |
then have "inv (x \<otimes> y) \<otimes> (x \<otimes> y) = (inv y \<otimes> inv x) \<otimes> (x \<otimes> y)" |
14963 | 375 |
by (simp add: m_assoc l_inv) (simp add: m_assoc [symmetric]) |
27698 | 376 |
with G show ?thesis by (simp del: l_inv Units_l_inv) |
13813 | 377 |
qed |
378 |
||
13940 | 379 |
lemma (in group) inv_comm: |
380 |
"[| x \<otimes> y = \<one>; x \<in> carrier G; y \<in> carrier G |] ==> y \<otimes> x = \<one>" |
|
14693 | 381 |
by (rule Units_inv_comm) auto |
13940 | 382 |
|
13944 | 383 |
lemma (in group) inv_equality: |
13943 | 384 |
"[|y \<otimes> x = \<one>; x \<in> carrier G; y \<in> carrier G|] ==> inv x = y" |
385 |
apply (simp add: m_inv_def) |
|
386 |
apply (rule the_equality) |
|
14693 | 387 |
apply (simp add: inv_comm [of y x]) |
388 |
apply (rule r_cancel [THEN iffD1], auto) |
|
13943 | 389 |
done |
390 |
||
13936 | 391 |
text {* Power *} |
392 |
||
393 |
lemma (in group) int_pow_def2: |
|
394 |
"a (^) (z::int) = (if neg z then inv (a (^) (nat (-z))) else a (^) (nat z))" |
|
395 |
by (simp add: int_pow_def nat_pow_def Let_def) |
|
396 |
||
397 |
lemma (in group) int_pow_0 [simp]: |
|
398 |
"x (^) (0::int) = \<one>" |
|
399 |
by (simp add: int_pow_def2) |
|
400 |
||
401 |
lemma (in group) int_pow_one [simp]: |
|
402 |
"\<one> (^) (z::int) = \<one>" |
|
403 |
by (simp add: int_pow_def2) |
|
404 |
||
20318
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
19984
diff
changeset
|
405 |
|
14963 | 406 |
subsection {* Subgroups *} |
13813 | 407 |
|
19783 | 408 |
locale subgroup = |
409 |
fixes H and G (structure) |
|
14963 | 410 |
assumes subset: "H \<subseteq> carrier G" |
411 |
and m_closed [intro, simp]: "\<lbrakk>x \<in> H; y \<in> H\<rbrakk> \<Longrightarrow> x \<otimes> y \<in> H" |
|
20318
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
19984
diff
changeset
|
412 |
and one_closed [simp]: "\<one> \<in> H" |
14963 | 413 |
and m_inv_closed [intro,simp]: "x \<in> H \<Longrightarrow> inv x \<in> H" |
13813 | 414 |
|
20318
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
19984
diff
changeset
|
415 |
lemma (in subgroup) is_subgroup: |
26199 | 416 |
"subgroup H G" by (rule subgroup_axioms) |
20318
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
19984
diff
changeset
|
417 |
|
13813 | 418 |
declare (in subgroup) group.intro [intro] |
13949
0ce528cd6f19
HOL-Algebra complete for release Isabelle2003 (modulo section headers).
ballarin
parents:
13944
diff
changeset
|
419 |
|
14963 | 420 |
lemma (in subgroup) mem_carrier [simp]: |
421 |
"x \<in> H \<Longrightarrow> x \<in> carrier G" |
|
422 |
using subset by blast |
|
13813 | 423 |
|
14963 | 424 |
lemma subgroup_imp_subset: |
425 |
"subgroup H G \<Longrightarrow> H \<subseteq> carrier G" |
|
426 |
by (rule subgroup.subset) |
|
427 |
||
428 |
lemma (in subgroup) subgroup_is_group [intro]: |
|
27611 | 429 |
assumes "group G" |
430 |
shows "group (G\<lparr>carrier := H\<rparr>)" |
|
431 |
proof - |
|
29237 | 432 |
interpret group G by fact |
27611 | 433 |
show ?thesis |
27698 | 434 |
apply (rule monoid.group_l_invI) |
435 |
apply (unfold_locales) [1] |
|
436 |
apply (auto intro: m_assoc l_inv mem_carrier) |
|
437 |
done |
|
27611 | 438 |
qed |
13813 | 439 |
|
440 |
text {* |
|
441 |
Since @{term H} is nonempty, it contains some element @{term x}. Since |
|
442 |
it is closed under inverse, it contains @{text "inv x"}. Since |
|
443 |
it is closed under product, it contains @{text "x \<otimes> inv x = \<one>"}. |
|
444 |
*} |
|
445 |
||
446 |
lemma (in group) one_in_subset: |
|
447 |
"[| H \<subseteq> carrier G; H \<noteq> {}; \<forall>a \<in> H. inv a \<in> H; \<forall>a\<in>H. \<forall>b\<in>H. a \<otimes> b \<in> H |] |
|
448 |
==> \<one> \<in> H" |
|
449 |
by (force simp add: l_inv) |
|
450 |
||
451 |
text {* A characterization of subgroups: closed, non-empty subset. *} |
|
452 |
||
453 |
lemma (in group) subgroupI: |
|
454 |
assumes subset: "H \<subseteq> carrier G" and non_empty: "H \<noteq> {}" |
|
14963 | 455 |
and inv: "!!a. a \<in> H \<Longrightarrow> inv a \<in> H" |
456 |
and mult: "!!a b. \<lbrakk>a \<in> H; b \<in> H\<rbrakk> \<Longrightarrow> a \<otimes> b \<in> H" |
|
13813 | 457 |
shows "subgroup H G" |
27714
27b4d7c01f8b
Tuned (for the sake of a meaningless log entry).
ballarin
parents:
27713
diff
changeset
|
458 |
proof (simp add: subgroup_def assms) |
27b4d7c01f8b
Tuned (for the sake of a meaningless log entry).
ballarin
parents:
27713
diff
changeset
|
459 |
show "\<one> \<in> H" by (rule one_in_subset) (auto simp only: assms) |
13813 | 460 |
qed |
461 |
||
13936 | 462 |
declare monoid.one_closed [iff] group.inv_closed [simp] |
463 |
monoid.l_one [simp] monoid.r_one [simp] group.inv_inv [simp] |
|
13813 | 464 |
|
465 |
lemma subgroup_nonempty: |
|
466 |
"~ subgroup {} G" |
|
467 |
by (blast dest: subgroup.one_closed) |
|
468 |
||
469 |
lemma (in subgroup) finite_imp_card_positive: |
|
470 |
"finite (carrier G) ==> 0 < card H" |
|
471 |
proof (rule classical) |
|
14963 | 472 |
assume "finite (carrier G)" "~ 0 < card H" |
473 |
then have "finite H" by (blast intro: finite_subset [OF subset]) |
|
474 |
with prems have "subgroup {} G" by simp |
|
13813 | 475 |
with subgroup_nonempty show ?thesis by contradiction |
476 |
qed |
|
477 |
||
13936 | 478 |
(* |
479 |
lemma (in monoid) Units_subgroup: |
|
480 |
"subgroup (Units G) G" |
|
481 |
*) |
|
482 |
||
20318
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
19984
diff
changeset
|
483 |
|
13813 | 484 |
subsection {* Direct Products *} |
485 |
||
35848
5443079512ea
slightly more uniform definitions -- eliminated old-style meta-equality;
wenzelm
parents:
35847
diff
changeset
|
486 |
definition |
5443079512ea
slightly more uniform definitions -- eliminated old-style meta-equality;
wenzelm
parents:
35847
diff
changeset
|
487 |
DirProd :: "_ \<Rightarrow> _ \<Rightarrow> ('a \<times> 'b) monoid" (infixr "\<times>\<times>" 80) where |
5443079512ea
slightly more uniform definitions -- eliminated old-style meta-equality;
wenzelm
parents:
35847
diff
changeset
|
488 |
"G \<times>\<times> H = |
5443079512ea
slightly more uniform definitions -- eliminated old-style meta-equality;
wenzelm
parents:
35847
diff
changeset
|
489 |
\<lparr>carrier = carrier G \<times> carrier H, |
5443079512ea
slightly more uniform definitions -- eliminated old-style meta-equality;
wenzelm
parents:
35847
diff
changeset
|
490 |
mult = (\<lambda>(g, h) (g', h'). (g \<otimes>\<^bsub>G\<^esub> g', h \<otimes>\<^bsub>H\<^esub> h')), |
5443079512ea
slightly more uniform definitions -- eliminated old-style meta-equality;
wenzelm
parents:
35847
diff
changeset
|
491 |
one = (\<one>\<^bsub>G\<^esub>, \<one>\<^bsub>H\<^esub>)\<rparr>" |
13813 | 492 |
|
14963 | 493 |
lemma DirProd_monoid: |
27611 | 494 |
assumes "monoid G" and "monoid H" |
14963 | 495 |
shows "monoid (G \<times>\<times> H)" |
496 |
proof - |
|
30729
461ee3e49ad3
interpretation/interpret: prefixes are mandatory by default;
wenzelm
parents:
29240
diff
changeset
|
497 |
interpret G: monoid G by fact |
461ee3e49ad3
interpretation/interpret: prefixes are mandatory by default;
wenzelm
parents:
29240
diff
changeset
|
498 |
interpret H: monoid H by fact |
27714
27b4d7c01f8b
Tuned (for the sake of a meaningless log entry).
ballarin
parents:
27713
diff
changeset
|
499 |
from assms |
14963 | 500 |
show ?thesis by (unfold monoid_def DirProd_def, auto) |
501 |
qed |
|
13813 | 502 |
|
503 |
||
14963 | 504 |
text{*Does not use the previous result because it's easier just to use auto.*} |
505 |
lemma DirProd_group: |
|
27611 | 506 |
assumes "group G" and "group H" |
14963 | 507 |
shows "group (G \<times>\<times> H)" |
27611 | 508 |
proof - |
30729
461ee3e49ad3
interpretation/interpret: prefixes are mandatory by default;
wenzelm
parents:
29240
diff
changeset
|
509 |
interpret G: group G by fact |
461ee3e49ad3
interpretation/interpret: prefixes are mandatory by default;
wenzelm
parents:
29240
diff
changeset
|
510 |
interpret H: group H by fact |
27611 | 511 |
show ?thesis by (rule groupI) |
14963 | 512 |
(auto intro: G.m_assoc H.m_assoc G.l_inv H.l_inv |
513 |
simp add: DirProd_def) |
|
27611 | 514 |
qed |
13813 | 515 |
|
14963 | 516 |
lemma carrier_DirProd [simp]: |
517 |
"carrier (G \<times>\<times> H) = carrier G \<times> carrier H" |
|
518 |
by (simp add: DirProd_def) |
|
13944 | 519 |
|
14963 | 520 |
lemma one_DirProd [simp]: |
521 |
"\<one>\<^bsub>G \<times>\<times> H\<^esub> = (\<one>\<^bsub>G\<^esub>, \<one>\<^bsub>H\<^esub>)" |
|
522 |
by (simp add: DirProd_def) |
|
13944 | 523 |
|
14963 | 524 |
lemma mult_DirProd [simp]: |
525 |
"(g, h) \<otimes>\<^bsub>(G \<times>\<times> H)\<^esub> (g', h') = (g \<otimes>\<^bsub>G\<^esub> g', h \<otimes>\<^bsub>H\<^esub> h')" |
|
526 |
by (simp add: DirProd_def) |
|
13944 | 527 |
|
14963 | 528 |
lemma inv_DirProd [simp]: |
27611 | 529 |
assumes "group G" and "group H" |
13944 | 530 |
assumes g: "g \<in> carrier G" |
531 |
and h: "h \<in> carrier H" |
|
14963 | 532 |
shows "m_inv (G \<times>\<times> H) (g, h) = (inv\<^bsub>G\<^esub> g, inv\<^bsub>H\<^esub> h)" |
27611 | 533 |
proof - |
30729
461ee3e49ad3
interpretation/interpret: prefixes are mandatory by default;
wenzelm
parents:
29240
diff
changeset
|
534 |
interpret G: group G by fact |
461ee3e49ad3
interpretation/interpret: prefixes are mandatory by default;
wenzelm
parents:
29240
diff
changeset
|
535 |
interpret H: group H by fact |
461ee3e49ad3
interpretation/interpret: prefixes are mandatory by default;
wenzelm
parents:
29240
diff
changeset
|
536 |
interpret Prod: group "G \<times>\<times> H" |
27714
27b4d7c01f8b
Tuned (for the sake of a meaningless log entry).
ballarin
parents:
27713
diff
changeset
|
537 |
by (auto intro: DirProd_group group.intro group.axioms assms) |
14963 | 538 |
show ?thesis by (simp add: Prod.inv_equality g h) |
539 |
qed |
|
27698 | 540 |
|
14963 | 541 |
|
542 |
subsection {* Homomorphisms and Isomorphisms *} |
|
13813 | 543 |
|
35847 | 544 |
definition |
545 |
hom :: "_ => _ => ('a => 'b) set" where |
|
35848
5443079512ea
slightly more uniform definitions -- eliminated old-style meta-equality;
wenzelm
parents:
35847
diff
changeset
|
546 |
"hom G H = |
13813 | 547 |
{h. h \<in> carrier G -> carrier H & |
14693 | 548 |
(\<forall>x \<in> carrier G. \<forall>y \<in> carrier G. h (x \<otimes>\<^bsub>G\<^esub> y) = h x \<otimes>\<^bsub>H\<^esub> h y)}" |
13813 | 549 |
|
14761 | 550 |
lemma (in group) hom_compose: |
31754 | 551 |
"[|h \<in> hom G H; i \<in> hom H I|] ==> compose (carrier G) i h \<in> hom G I" |
552 |
by (fastsimp simp add: hom_def compose_def) |
|
13943 | 553 |
|
35848
5443079512ea
slightly more uniform definitions -- eliminated old-style meta-equality;
wenzelm
parents:
35847
diff
changeset
|
554 |
definition |
5443079512ea
slightly more uniform definitions -- eliminated old-style meta-equality;
wenzelm
parents:
35847
diff
changeset
|
555 |
iso :: "_ => _ => ('a => 'b) set" (infixr "\<cong>" 60) |
5443079512ea
slightly more uniform definitions -- eliminated old-style meta-equality;
wenzelm
parents:
35847
diff
changeset
|
556 |
where "G \<cong> H = {h. h \<in> hom G H & bij_betw h (carrier G) (carrier H)}" |
14761 | 557 |
|
14803 | 558 |
lemma iso_refl: "(%x. x) \<in> G \<cong> G" |
31727 | 559 |
by (simp add: iso_def hom_def inj_on_def bij_betw_def Pi_def) |
14761 | 560 |
|
561 |
lemma (in group) iso_sym: |
|
33057 | 562 |
"h \<in> G \<cong> H \<Longrightarrow> inv_into (carrier G) h \<in> H \<cong> G" |
563 |
apply (simp add: iso_def bij_betw_inv_into) |
|
564 |
apply (subgoal_tac "inv_into (carrier G) h \<in> carrier H \<rightarrow> carrier G") |
|
565 |
prefer 2 apply (simp add: bij_betw_imp_funcset [OF bij_betw_inv_into]) |
|
566 |
apply (simp add: hom_def bij_betw_def inv_into_f_eq f_inv_into_f Pi_def) |
|
14761 | 567 |
done |
568 |
||
569 |
lemma (in group) iso_trans: |
|
14803 | 570 |
"[|h \<in> G \<cong> H; i \<in> H \<cong> I|] ==> (compose (carrier G) i h) \<in> G \<cong> I" |
14761 | 571 |
by (auto simp add: iso_def hom_compose bij_betw_compose) |
572 |
||
14963 | 573 |
lemma DirProd_commute_iso: |
574 |
shows "(\<lambda>(x,y). (y,x)) \<in> (G \<times>\<times> H) \<cong> (H \<times>\<times> G)" |
|
31754 | 575 |
by (auto simp add: iso_def hom_def inj_on_def bij_betw_def) |
14761 | 576 |
|
14963 | 577 |
lemma DirProd_assoc_iso: |
578 |
shows "(\<lambda>(x,y,z). (x,(y,z))) \<in> (G \<times>\<times> H \<times>\<times> I) \<cong> (G \<times>\<times> (H \<times>\<times> I))" |
|
31727 | 579 |
by (auto simp add: iso_def hom_def inj_on_def bij_betw_def) |
14761 | 580 |
|
581 |
||
14963 | 582 |
text{*Basis for homomorphism proofs: we assume two groups @{term G} and |
15076
4b3d280ef06a
New prover for transitive and reflexive-transitive closure of relations.
ballarin
parents:
14963
diff
changeset
|
583 |
@{term H}, with a homomorphism @{term h} between them*} |
29237 | 584 |
locale group_hom = G: group G + H: group H for G (structure) and H (structure) + |
585 |
fixes h |
|
13813 | 586 |
assumes homh: "h \<in> hom G H" |
29240 | 587 |
|
588 |
lemma (in group_hom) hom_mult [simp]: |
|
589 |
"[| x \<in> carrier G; y \<in> carrier G |] ==> h (x \<otimes>\<^bsub>G\<^esub> y) = h x \<otimes>\<^bsub>H\<^esub> h y" |
|
590 |
proof - |
|
591 |
assume "x \<in> carrier G" "y \<in> carrier G" |
|
592 |
with homh [unfolded hom_def] show ?thesis by simp |
|
593 |
qed |
|
594 |
||
595 |
lemma (in group_hom) hom_closed [simp]: |
|
596 |
"x \<in> carrier G ==> h x \<in> carrier H" |
|
597 |
proof - |
|
598 |
assume "x \<in> carrier G" |
|
31754 | 599 |
with homh [unfolded hom_def] show ?thesis by auto |
29240 | 600 |
qed |
13813 | 601 |
|
602 |
lemma (in group_hom) one_closed [simp]: |
|
603 |
"h \<one> \<in> carrier H" |
|
604 |
by simp |
|
605 |
||
606 |
lemma (in group_hom) hom_one [simp]: |
|
14693 | 607 |
"h \<one> = \<one>\<^bsub>H\<^esub>" |
13813 | 608 |
proof - |
15076
4b3d280ef06a
New prover for transitive and reflexive-transitive closure of relations.
ballarin
parents:
14963
diff
changeset
|
609 |
have "h \<one> \<otimes>\<^bsub>H\<^esub> \<one>\<^bsub>H\<^esub> = h \<one> \<otimes>\<^bsub>H\<^esub> h \<one>" |
13813 | 610 |
by (simp add: hom_mult [symmetric] del: hom_mult) |
611 |
then show ?thesis by (simp del: r_one) |
|
612 |
qed |
|
613 |
||
614 |
lemma (in group_hom) inv_closed [simp]: |
|
615 |
"x \<in> carrier G ==> h (inv x) \<in> carrier H" |
|
616 |
by simp |
|
617 |
||
618 |
lemma (in group_hom) hom_inv [simp]: |
|
14693 | 619 |
"x \<in> carrier G ==> h (inv x) = inv\<^bsub>H\<^esub> (h x)" |
13813 | 620 |
proof - |
621 |
assume x: "x \<in> carrier G" |
|
14693 | 622 |
then have "h x \<otimes>\<^bsub>H\<^esub> h (inv x) = \<one>\<^bsub>H\<^esub>" |
14963 | 623 |
by (simp add: hom_mult [symmetric] del: hom_mult) |
14693 | 624 |
also from x have "... = h x \<otimes>\<^bsub>H\<^esub> inv\<^bsub>H\<^esub> (h x)" |
14963 | 625 |
by (simp add: hom_mult [symmetric] del: hom_mult) |
14693 | 626 |
finally have "h x \<otimes>\<^bsub>H\<^esub> h (inv x) = h x \<otimes>\<^bsub>H\<^esub> inv\<^bsub>H\<^esub> (h x)" . |
27698 | 627 |
with x show ?thesis by (simp del: H.r_inv H.Units_r_inv) |
13813 | 628 |
qed |
629 |
||
20318
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
19984
diff
changeset
|
630 |
|
13949
0ce528cd6f19
HOL-Algebra complete for release Isabelle2003 (modulo section headers).
ballarin
parents:
13944
diff
changeset
|
631 |
subsection {* Commutative Structures *} |
13936 | 632 |
|
633 |
text {* |
|
634 |
Naming convention: multiplicative structures that are commutative |
|
635 |
are called \emph{commutative}, additive structures are called |
|
636 |
\emph{Abelian}. |
|
637 |
*} |
|
13813 | 638 |
|
14963 | 639 |
locale comm_monoid = monoid + |
640 |
assumes m_comm: "\<lbrakk>x \<in> carrier G; y \<in> carrier G\<rbrakk> \<Longrightarrow> x \<otimes> y = y \<otimes> x" |
|
13813 | 641 |
|
14963 | 642 |
lemma (in comm_monoid) m_lcomm: |
643 |
"\<lbrakk>x \<in> carrier G; y \<in> carrier G; z \<in> carrier G\<rbrakk> \<Longrightarrow> |
|
13813 | 644 |
x \<otimes> (y \<otimes> z) = y \<otimes> (x \<otimes> z)" |
645 |
proof - |
|
14693 | 646 |
assume xyz: "x \<in> carrier G" "y \<in> carrier G" "z \<in> carrier G" |
13813 | 647 |
from xyz have "x \<otimes> (y \<otimes> z) = (x \<otimes> y) \<otimes> z" by (simp add: m_assoc) |
648 |
also from xyz have "... = (y \<otimes> x) \<otimes> z" by (simp add: m_comm) |
|
649 |
also from xyz have "... = y \<otimes> (x \<otimes> z)" by (simp add: m_assoc) |
|
650 |
finally show ?thesis . |
|
651 |
qed |
|
652 |
||
14963 | 653 |
lemmas (in comm_monoid) m_ac = m_assoc m_comm m_lcomm |
13813 | 654 |
|
13936 | 655 |
lemma comm_monoidI: |
19783 | 656 |
fixes G (structure) |
13936 | 657 |
assumes m_closed: |
14693 | 658 |
"!!x y. [| x \<in> carrier G; y \<in> carrier G |] ==> x \<otimes> y \<in> carrier G" |
659 |
and one_closed: "\<one> \<in> carrier G" |
|
13936 | 660 |
and m_assoc: |
661 |
"!!x y z. [| x \<in> carrier G; y \<in> carrier G; z \<in> carrier G |] ==> |
|
14693 | 662 |
(x \<otimes> y) \<otimes> z = x \<otimes> (y \<otimes> z)" |
663 |
and l_one: "!!x. x \<in> carrier G ==> \<one> \<otimes> x = x" |
|
13936 | 664 |
and m_comm: |
14693 | 665 |
"!!x y. [| x \<in> carrier G; y \<in> carrier G |] ==> x \<otimes> y = y \<otimes> x" |
13936 | 666 |
shows "comm_monoid G" |
667 |
using l_one |
|
14963 | 668 |
by (auto intro!: comm_monoid.intro comm_monoid_axioms.intro monoid.intro |
27714
27b4d7c01f8b
Tuned (for the sake of a meaningless log entry).
ballarin
parents:
27713
diff
changeset
|
669 |
intro: assms simp: m_closed one_closed m_comm) |
13817 | 670 |
|
13936 | 671 |
lemma (in monoid) monoid_comm_monoidI: |
672 |
assumes m_comm: |
|
14693 | 673 |
"!!x y. [| x \<in> carrier G; y \<in> carrier G |] ==> x \<otimes> y = y \<otimes> x" |
13936 | 674 |
shows "comm_monoid G" |
675 |
by (rule comm_monoidI) (auto intro: m_assoc m_comm) |
|
14963 | 676 |
|
14693 | 677 |
(*lemma (in comm_monoid) r_one [simp]: |
13817 | 678 |
"x \<in> carrier G ==> x \<otimes> \<one> = x" |
679 |
proof - |
|
680 |
assume G: "x \<in> carrier G" |
|
681 |
then have "x \<otimes> \<one> = \<one> \<otimes> x" by (simp add: m_comm) |
|
682 |
also from G have "... = x" by simp |
|
683 |
finally show ?thesis . |
|
14693 | 684 |
qed*) |
14963 | 685 |
|
13936 | 686 |
lemma (in comm_monoid) nat_pow_distr: |
687 |
"[| x \<in> carrier G; y \<in> carrier G |] ==> |
|
688 |
(x \<otimes> y) (^) (n::nat) = x (^) n \<otimes> y (^) n" |
|
689 |
by (induct n) (simp, simp add: m_ac) |
|
690 |
||
691 |
locale comm_group = comm_monoid + group |
|
692 |
||
693 |
lemma (in group) group_comm_groupI: |
|
694 |
assumes m_comm: "!!x y. [| x \<in> carrier G; y \<in> carrier G |] ==> |
|
14693 | 695 |
x \<otimes> y = y \<otimes> x" |
13936 | 696 |
shows "comm_group G" |
28823 | 697 |
proof qed (simp_all add: m_comm) |
13817 | 698 |
|
13936 | 699 |
lemma comm_groupI: |
19783 | 700 |
fixes G (structure) |
13936 | 701 |
assumes m_closed: |
14693 | 702 |
"!!x y. [| x \<in> carrier G; y \<in> carrier G |] ==> x \<otimes> y \<in> carrier G" |
703 |
and one_closed: "\<one> \<in> carrier G" |
|
13936 | 704 |
and m_assoc: |
705 |
"!!x y z. [| x \<in> carrier G; y \<in> carrier G; z \<in> carrier G |] ==> |
|
14693 | 706 |
(x \<otimes> y) \<otimes> z = x \<otimes> (y \<otimes> z)" |
13936 | 707 |
and m_comm: |
14693 | 708 |
"!!x y. [| x \<in> carrier G; y \<in> carrier G |] ==> x \<otimes> y = y \<otimes> x" |
709 |
and l_one: "!!x. x \<in> carrier G ==> \<one> \<otimes> x = x" |
|
14963 | 710 |
and l_inv_ex: "!!x. x \<in> carrier G ==> \<exists>y \<in> carrier G. y \<otimes> x = \<one>" |
13936 | 711 |
shows "comm_group G" |
27714
27b4d7c01f8b
Tuned (for the sake of a meaningless log entry).
ballarin
parents:
27713
diff
changeset
|
712 |
by (fast intro: group.group_comm_groupI groupI assms) |
13936 | 713 |
|
714 |
lemma (in comm_group) inv_mult: |
|
13854
91c9ab25fece
First distributed version of Group and Ring theory.
ballarin
parents:
13835
diff
changeset
|
715 |
"[| x \<in> carrier G; y \<in> carrier G |] ==> inv (x \<otimes> y) = inv x \<otimes> inv y" |
13936 | 716 |
by (simp add: m_ac inv_mult_group) |
13854
91c9ab25fece
First distributed version of Group and Ring theory.
ballarin
parents:
13835
diff
changeset
|
717 |
|
20318
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
19984
diff
changeset
|
718 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
19984
diff
changeset
|
719 |
subsection {* The Lattice of Subgroups of a Group *} |
14751
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
720 |
|
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
721 |
text_raw {* \label{sec:subgroup-lattice} *} |
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
722 |
|
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
723 |
theorem (in group) subgroups_partial_order: |
27713
95b36bfe7fc4
New locales for orders and lattices where the equivalence relation is not restricted to equality.
ballarin
parents:
27698
diff
changeset
|
724 |
"partial_order (| carrier = {H. subgroup H G}, eq = op =, le = op \<subseteq> |)" |
28823 | 725 |
proof qed simp_all |
14751
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
726 |
|
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
727 |
lemma (in group) subgroup_self: |
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
728 |
"subgroup (carrier G) G" |
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
729 |
by (rule subgroupI) auto |
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
730 |
|
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
731 |
lemma (in group) subgroup_imp_group: |
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
732 |
"subgroup H G ==> group (G(| carrier := H |))" |
26199 | 733 |
by (erule subgroup.subgroup_is_group) (rule group_axioms) |
14751
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
734 |
|
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
735 |
lemma (in group) is_monoid [intro, simp]: |
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
736 |
"monoid G" |
14963 | 737 |
by (auto intro: monoid.intro m_assoc) |
14751
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
738 |
|
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
739 |
lemma (in group) subgroup_inv_equality: |
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
740 |
"[| subgroup H G; x \<in> H |] ==> m_inv (G (| carrier := H |)) x = inv x" |
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
741 |
apply (rule_tac inv_equality [THEN sym]) |
14761 | 742 |
apply (rule group.l_inv [OF subgroup_imp_group, simplified], assumption+) |
743 |
apply (rule subsetD [OF subgroup.subset], assumption+) |
|
744 |
apply (rule subsetD [OF subgroup.subset], assumption) |
|
745 |
apply (rule_tac group.inv_closed [OF subgroup_imp_group, simplified], assumption+) |
|
14751
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
746 |
done |
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
747 |
|
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
748 |
theorem (in group) subgroups_Inter: |
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
749 |
assumes subgr: "(!!H. H \<in> A ==> subgroup H G)" |
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
750 |
and not_empty: "A ~= {}" |
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
751 |
shows "subgroup (\<Inter>A) G" |
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
752 |
proof (rule subgroupI) |
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
753 |
from subgr [THEN subgroup.subset] and not_empty |
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
754 |
show "\<Inter>A \<subseteq> carrier G" by blast |
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
755 |
next |
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
756 |
from subgr [THEN subgroup.one_closed] |
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
757 |
show "\<Inter>A ~= {}" by blast |
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
758 |
next |
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
759 |
fix x assume "x \<in> \<Inter>A" |
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
760 |
with subgr [THEN subgroup.m_inv_closed] |
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
761 |
show "inv x \<in> \<Inter>A" by blast |
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
762 |
next |
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
763 |
fix x y assume "x \<in> \<Inter>A" "y \<in> \<Inter>A" |
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
764 |
with subgr [THEN subgroup.m_closed] |
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
765 |
show "x \<otimes> y \<in> \<Inter>A" by blast |
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
766 |
qed |
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
767 |
|
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
768 |
theorem (in group) subgroups_complete_lattice: |
27713
95b36bfe7fc4
New locales for orders and lattices where the equivalence relation is not restricted to equality.
ballarin
parents:
27698
diff
changeset
|
769 |
"complete_lattice (| carrier = {H. subgroup H G}, eq = op =, le = op \<subseteq> |)" |
22063
717425609192
Reverted to structure representation with records.
ballarin
parents:
21041
diff
changeset
|
770 |
(is "complete_lattice ?L") |
14751
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
771 |
proof (rule partial_order.complete_lattice_criterion1) |
22063
717425609192
Reverted to structure representation with records.
ballarin
parents:
21041
diff
changeset
|
772 |
show "partial_order ?L" by (rule subgroups_partial_order) |
14751
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
773 |
next |
26805
27941d7d9a11
Replaced forward proofs of existential statements by backward proofs
berghofe
parents:
26199
diff
changeset
|
774 |
show "\<exists>G. greatest ?L G (carrier ?L)" |
27941d7d9a11
Replaced forward proofs of existential statements by backward proofs
berghofe
parents:
26199
diff
changeset
|
775 |
proof |
27941d7d9a11
Replaced forward proofs of existential statements by backward proofs
berghofe
parents:
26199
diff
changeset
|
776 |
show "greatest ?L (carrier G) (carrier ?L)" |
27941d7d9a11
Replaced forward proofs of existential statements by backward proofs
berghofe
parents:
26199
diff
changeset
|
777 |
by (unfold greatest_def) |
27941d7d9a11
Replaced forward proofs of existential statements by backward proofs
berghofe
parents:
26199
diff
changeset
|
778 |
(simp add: subgroup.subset subgroup_self) |
27941d7d9a11
Replaced forward proofs of existential statements by backward proofs
berghofe
parents:
26199
diff
changeset
|
779 |
qed |
14751
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
780 |
next |
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
781 |
fix A |
22063
717425609192
Reverted to structure representation with records.
ballarin
parents:
21041
diff
changeset
|
782 |
assume L: "A \<subseteq> carrier ?L" and non_empty: "A ~= {}" |
14751
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
783 |
then have Int_subgroup: "subgroup (\<Inter>A) G" |
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
784 |
by (fastsimp intro: subgroups_Inter) |
26805
27941d7d9a11
Replaced forward proofs of existential statements by backward proofs
berghofe
parents:
26199
diff
changeset
|
785 |
show "\<exists>I. greatest ?L I (Lower ?L A)" |
27941d7d9a11
Replaced forward proofs of existential statements by backward proofs
berghofe
parents:
26199
diff
changeset
|
786 |
proof |
27941d7d9a11
Replaced forward proofs of existential statements by backward proofs
berghofe
parents:
26199
diff
changeset
|
787 |
show "greatest ?L (\<Inter>A) (Lower ?L A)" |
27941d7d9a11
Replaced forward proofs of existential statements by backward proofs
berghofe
parents:
26199
diff
changeset
|
788 |
(is "greatest _ ?Int _") |
27941d7d9a11
Replaced forward proofs of existential statements by backward proofs
berghofe
parents:
26199
diff
changeset
|
789 |
proof (rule greatest_LowerI) |
27941d7d9a11
Replaced forward proofs of existential statements by backward proofs
berghofe
parents:
26199
diff
changeset
|
790 |
fix H |
27941d7d9a11
Replaced forward proofs of existential statements by backward proofs
berghofe
parents:
26199
diff
changeset
|
791 |
assume H: "H \<in> A" |
27941d7d9a11
Replaced forward proofs of existential statements by backward proofs
berghofe
parents:
26199
diff
changeset
|
792 |
with L have subgroupH: "subgroup H G" by auto |
27941d7d9a11
Replaced forward proofs of existential statements by backward proofs
berghofe
parents:
26199
diff
changeset
|
793 |
from subgroupH have groupH: "group (G (| carrier := H |))" (is "group ?H") |
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
31754
diff
changeset
|
794 |
by (rule subgroup_imp_group) |
26805
27941d7d9a11
Replaced forward proofs of existential statements by backward proofs
berghofe
parents:
26199
diff
changeset
|
795 |
from groupH have monoidH: "monoid ?H" |
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
31754
diff
changeset
|
796 |
by (rule group.is_monoid) |
26805
27941d7d9a11
Replaced forward proofs of existential statements by backward proofs
berghofe
parents:
26199
diff
changeset
|
797 |
from H have Int_subset: "?Int \<subseteq> H" by fastsimp |
27941d7d9a11
Replaced forward proofs of existential statements by backward proofs
berghofe
parents:
26199
diff
changeset
|
798 |
then show "le ?L ?Int H" by simp |
27941d7d9a11
Replaced forward proofs of existential statements by backward proofs
berghofe
parents:
26199
diff
changeset
|
799 |
next |
27941d7d9a11
Replaced forward proofs of existential statements by backward proofs
berghofe
parents:
26199
diff
changeset
|
800 |
fix H |
27941d7d9a11
Replaced forward proofs of existential statements by backward proofs
berghofe
parents:
26199
diff
changeset
|
801 |
assume H: "H \<in> Lower ?L A" |
27941d7d9a11
Replaced forward proofs of existential statements by backward proofs
berghofe
parents:
26199
diff
changeset
|
802 |
with L Int_subgroup show "le ?L H ?Int" |
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
31754
diff
changeset
|
803 |
by (fastsimp simp: Lower_def intro: Inter_greatest) |
26805
27941d7d9a11
Replaced forward proofs of existential statements by backward proofs
berghofe
parents:
26199
diff
changeset
|
804 |
next |
27941d7d9a11
Replaced forward proofs of existential statements by backward proofs
berghofe
parents:
26199
diff
changeset
|
805 |
show "A \<subseteq> carrier ?L" by (rule L) |
27941d7d9a11
Replaced forward proofs of existential statements by backward proofs
berghofe
parents:
26199
diff
changeset
|
806 |
next |
27941d7d9a11
Replaced forward proofs of existential statements by backward proofs
berghofe
parents:
26199
diff
changeset
|
807 |
show "?Int \<in> carrier ?L" by simp (rule Int_subgroup) |
27941d7d9a11
Replaced forward proofs of existential statements by backward proofs
berghofe
parents:
26199
diff
changeset
|
808 |
qed |
14751
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
809 |
qed |
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
810 |
qed |
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
811 |
|
13813 | 812 |
end |